Further Trigonometric Equations

Summary

What Makes a Trigonometric Equation "Hard"?

When you first encounter trig equations, you might solve something like sin x = 0.5 and find two answers: 30° and 150° (in the range 0° to 360°). That feels manageable.

But a harder trig equation might look like:

Why is this harder? Because it contains:

• Reciprocal functions (cot, sec, cosec) that aren't in a simple form
• Multiple trig functions (cos and cot together)
• Products or sums that need to be expanded or rearranged
• Quadratic-like structures that require factorising
• Different multiples of the variable (like sin 2x and sin x in the same equation)

The Systematic Strategy

Think of solving hard trig equations as a decision tree. At each step, you ask a question and follow a path. Let me break this down into manageable chunks:

Step 1: Do you have different multiples of x or θ?

Examples of different multiples:

• sin 2x and sin x in the same equation
• cos 3θ and cos θ together
• tan x and tan 2x mixed

If YES: Use double angle formulas (for 2x) or compound angle formulas (for addition/subtraction) to express everything in terms of the same multiple. For example, sin 2x = 2 sin x cos x.

If NO: Move to Step 2.

Step 2: Does the equation contain a function of x or θ?

By "function of x or θ," I mean the variable itself is being transformed. Examples:

• sin 2x (the argument is 2x, not just x)
• cos(θ − 60°) (the argument is θ − 60°, not just θ)
• tan(x/2) (the argument is x/2, not just x)

If YES: Transform your range FIRST before solving. For example, if you're solving sin 2x for 0° ≤ x < 360°, the range for 2x is 0° ≤ 2x < 720°. Solve for 2x in this transformed range, then divide your final answers by 2 to get x.

If NO: Keep your range as is and move to Step 3.

Dealing with Reciprocal & Inverse Functions